Abstract

The Zaneveld–Wells algorithm for calculating N inherent optical expansion coefficients from N + 1 measured angle-integrated moments of the radiant light field is investigated. Because the algorithm is well conditioned but sensitive to errors in the spatial derivatives, different approximations for the spatial derivatives are considered. The effects of noise and sensor error on the performance of the algorithm have been evaluated analytically, and testing with randomly sampled simulated noise was performed to assess the stability and sensitivity of the algorithm. Results show that the algorithm is fairly insensitive to sensor noise, but neither using a higher-order finite-difference approximation for the derivatives nor reformulating the algorithm into an integral form was successful in overcoming the large errors observed.

References

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Table 4

Percent Error in Calculated An Coefficients at 570 nm as a Function of the Derivative Approximation Order and the Chlorophyll Concentration (mg/m3) at an Optical Depth of 10

n of An

First-Order Difference Spatial Derivative

Fourth-Order Difference Spatial Derivative

C = 0

C = 0.05

C = 0.1

C = 0.5

C = 1.0

C = 0

C = 0.05

C = 0.1

C = 0.5

C = 1.0

0

−4.45

−1.57

1.07

7.88

9.55

0.02

3.77

6.44

13.41

15.46

1

−4.50

21.36

4.47

59.44

16.48

−0.04

25.49

9.65

61.88

21.93

2

−4.32

10.53

4.79

37.04

18.32

0.09

15.22

9.95

40.83

23.67

3

−4.41

2.79

−22.00

21.46

−50.97

−0.05

7.87

−15.40

26.19

−41.04

4

−4.08

−10.08

−18.95

−9.90

−43.56

0.19

−4.38

−12.54

−3.22

−33.90

5

−4.23

2.67

−16.46

20.00

−33.62

−0.04

7.66

−10.22

25.01

−23.96

6

−3.28

−5.47

−10.34

19.53

25.09

0.76

−0.15

−4.47

24.91

32.54

7

−3.66

4.90

−6.39

46.09

39.08

0.28

9.63

−0.73

50.01

44.59

8

0.90

−0.74

−3.77

7.75

−14.96

4.56

4.21

1.81

15.11

−3.34

Table 5

Percent Error in Calculated AnCoefficients at 570 nm with the Integral Form of the Algorithm as a Function of the Number of Measurements in Depth Used and the Chlorophyll Concentration (mg/m3) at an Optical Depth of 10

n of An

Integration between Two Measurements

Integration over Four Measurements

C = 0

C = 0.05

C = 0.1

C = 0.5

C = 1.0

C = 0

C = 0.05

C = 0.1

C = 0.5

C = 1.0

0

0.08

3.85

6.53

13.52

15.58

0.02

3.76

6.43

13.41

15.45

1

0.02

25.56

9.74

61.93

22.04

−0.04

25.49

9.65

61.88

21.93

2

0.15

15.30

10.04

40.90

23.78

0.09

15.22

9.95

40.82

23.66

3

0.01

7.95

−15.29

26.28

−40.84

−0.05

7.86

−15.41

26.19

−41.05

4

0.25

−4.29

−12.44

−3.09

−33.69

0.19

−4.39

−12.55

−3.22

−33.87

5

0.01

7.74

−10.12

25.10

−23.74

−0.04

7.65

−10.23

25.00

−23.94

6

0.81

−0.06

−4.36

24.90

32.17

0.75

−0.14

−4.45

24.38

30.15

7

0.33

9.71

−0.62

49.87

43.96

0.29

9.66

−0.69

49.02

41.06

8

4.60

4.30

1.93

15.24

−2.82

4.53

4.27

1.92

14.90

−2.63

Tables (5)

Table 1

Theoretical Optical Properties for the Numerical Tests at 570 nm

C (mg/m3)

c (m−1)

b/c

K∞ (m−1)

Cutoff Angle (deg)

Forward Fraction

N of fn

0

0.0816

0.0208

0.0816

2

0.05

0.1295

0.3620

0.110

1.5489

0.1090

90

0.10

0.1552

0.4584

0.120

1.9953

0.1448

90

0.50

0.2809

0.6767

0.157

2.5119

0.2022

80

1.00

0.3876

0.7512

0.181

2.5119

0.2099

80

Table 2

Percent Error in the Calculated Values of An for Various Nondimensional Distances between Measurements with a First-Order Finite-Difference Scheme at 570 nm in the Asymptotic Regimea

n of An

True An

Δτ = 0.001

Δτ = 0.100

Δτ = 0.500

0

0.5416

17.05

14.02

0.16

1

0.7022

2.01

−1.58

−17.95

2

0.7091

−0.46

−4.13

−20.92

3

0.7533

−1.37

−5.08

−22.02

4

0.7740

−1.60

−5.32

−22.29

5

0.7920

−2.19

−5.94

−23.01

6

0.8073

−1.78

−5.50

−22.51

7

0.8210

−3.07

−6.84

−24.06

8

0.8329

0.66

−2.97

−19.57

aC= 0.1 mg/m3, c = 0.1552 m−1.

Table 3

Percent Error in the Calculated Values of An for Various Nondimensional Distances between Measurements with a First-Order Finite-Difference Scheme at 570 nm in the Asymptotic Regimea

Table 4

Percent Error in Calculated An Coefficients at 570 nm as a Function of the Derivative Approximation Order and the Chlorophyll Concentration (mg/m3) at an Optical Depth of 10

n of An

First-Order Difference Spatial Derivative

Fourth-Order Difference Spatial Derivative

C = 0

C = 0.05

C = 0.1

C = 0.5

C = 1.0

C = 0

C = 0.05

C = 0.1

C = 0.5

C = 1.0

0

−4.45

−1.57

1.07

7.88

9.55

0.02

3.77

6.44

13.41

15.46

1

−4.50

21.36

4.47

59.44

16.48

−0.04

25.49

9.65

61.88

21.93

2

−4.32

10.53

4.79

37.04

18.32

0.09

15.22

9.95

40.83

23.67

3

−4.41

2.79

−22.00

21.46

−50.97

−0.05

7.87

−15.40

26.19

−41.04

4

−4.08

−10.08

−18.95

−9.90

−43.56

0.19

−4.38

−12.54

−3.22

−33.90

5

−4.23

2.67

−16.46

20.00

−33.62

−0.04

7.66

−10.22

25.01

−23.96

6

−3.28

−5.47

−10.34

19.53

25.09

0.76

−0.15

−4.47

24.91

32.54

7

−3.66

4.90

−6.39

46.09

39.08

0.28

9.63

−0.73

50.01

44.59

8

0.90

−0.74

−3.77

7.75

−14.96

4.56

4.21

1.81

15.11

−3.34

Table 5

Percent Error in Calculated AnCoefficients at 570 nm with the Integral Form of the Algorithm as a Function of the Number of Measurements in Depth Used and the Chlorophyll Concentration (mg/m3) at an Optical Depth of 10